Probability Lab
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The 68-95-99.7 Rule Explained

Why These Specific Percentages?

The 68-95-99.7 rule (also called the Empirical Rule) describes how data is distributed in a normal distribution. These aren't arbitrary numbers – they correspond to exactly 1, 2, and 3 standard deviations from the mean.

±1σ (68.27%)

About 68% of data falls within one standard deviation of the mean.

\[P(\mu - \sigma < X < \mu + \sigma) \approx 0.6827\]

CI: Use z = 1.00 for ~68% confidence

±2σ (95.45%)

About 95% of data falls within two standard deviations of the mean.

\[P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.9545\]

CI: Use z = 1.96 for exactly 95% confidence

±3σ (99.73%)

About 99.7% of data falls within three standard deviations of the mean.

\[P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.9973\]

CI: Use z = 2.58 for exactly 99% confidence

Common Confidence Levels

Standard Levels:

  • • 90% confidence: z = 1.645
  • • 95% confidence: z = 1.960
  • • 99% confidence: z = 2.576

Why Not Exactly 2?

While ±2σ gives 95.45%, we often want exactly 95%. That's why we use z = 1.96 instead of 2.00 for 95% CIs.

Connection to Quality Control

The 68-95-99.7 rule is fundamental in Six Sigma quality control:

  • • 3σ control limits catch 99.73% of normal variation
  • • Anything outside ±3σ is likely a special cause
  • • "Six Sigma" means the specification limits are ±6σ from the mean